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In this paper, we investigate the monotonicity pattern of the function x → lnΓ(x+1) ln (x2 +a)− ln (x+a) on (0,1) for a 1 and resolve an open problem. From which we prove that the double inequality ( x2 +a x+a )(1−γ)(a+1) < Γ(x+1) < ( x2 +b x+b )(1−γ)(b+1) holds for x∈ (0,1) if and only if 0 < a (1− γ)/(2γ −1) and b (π2 −6γ) / (18−12γ −π2) , while the double inequality ( x2 +a x+a )γa < Γ(x+1) < ( x2 +b x+b )γb holds for x ∈ (0,1) if and only if a (1− γ)/(2γ −1) and 0 < b 6γ/(π2 − 12γ) , where γ = 0.577 . . . denotes Euler-Mascheroni’s constant. These greatly improve some existing results.

Monotonicity and sharp inequalities related to gamma function